Ginormous
by Derek Kisman

Hmm.  What to do with a 1364-digit number?  Well, it does end in a 0.
Perhaps factorizing it is a good idea.  Indeed, this puzzle turns out to be all
about playing with prime factors of big numbers.  Some software to factorize large
integers (into small prime factors) would be helpful; it's easy to write
such a program for yourself, or you can use plenty of advanced math programs.

The 1364-digit number factors into the first 129 primes as follows:
2^1 * 3^4 * 5^2 * 7^9 * 11^9 * 13^4 * 17^5 * 19^9 * 23^2 * 29^8 * 31^7 *
37^1 * 41^0 * 43^8 * 47^0 * 53^7 * 59^7 * 61^6 * 67^6 * 71^6 * 73^5 * 79^3 * 83^6 *
89^7 * 97^3 * 101^7 * 103^7 * 107^0 * 109^1 * 113^0 * 127^3 * 131^3 * 137^0
* 139^9 * 149^7 * 151^5 * 157^6 * 163^0 * 167^9 * 173^3 * 179^4 * 181^2 *
191^9 * 193^1 * 197^1 * 199^5 * 211^9 * 223^0 * 227^9 * 229^4 * 233^7 * 239^4 *
241^9 * 251^3 * 257^6 * 263^7 * 269^2 * 271^5 * 277^1 * 281^0 * 283^9 * 293^2 *
307^3 * 311^9 * 313^8 * 317^0 * 331^2 * 337^2 * 347^6 * 349^3 * 353^4 * 359^5 *
367^6 * 373^5 * 379^0 * 383^3 * 389^6 * 397^8 * 401^0 * 409^9 * 419^5 * 421^5 *
431^2 * 433^9 * 439^4 * 443^9 * 449^7 * 457^3 * 461^0 * 463^6 * 467^3 * 479^2 *
487^3 * 491^2 * 499^6 * 503^5 * 509^2 * 521^3 * 523^4 * 541^4 * 547^5 * 557^1 *
563^5 * 569^8 * 571^8 * 577^2 * 587^7 * 593^3 * 599^6 * 601^2 * 607^8 * 613^4 *
617^9 * 619^2 * 631^0 * 641^1 * 643^8 * 647^4 * 653^1 * 659^3 * 661^5 * 673^7 *
677^9 * 683^7 * 691^0 * 701^2 * 709^5 * 719^8 * 727^9

Notice how every exponent is between 0 and 9?  This suggests that we can
take the exponents and make them into a new number.  When we do that, we get:
142994592871080776665367377010330975609342911590947493672510923980226345650368095529497306323265234451588273628492018413579702589

This has another interesting prime factorization:
13 * 47 * 67 * 79 * 97 * 127 * 149 * 181 * 197 * 251 * 281 * 307 * 367 * 373
* 379 * 401 * 431 * 487 * 557 * 593 * 643 * 701 * 751 * 821 * 823 * 863 * 883
* 929 * 937 * 997 * 1009 * 1061 * 1117 * 1153 * 1213 * 1237 * 1259 * 1303 *
1361 * 1399 * 1453 * 1483 * 1489 * 1559 * 1597 * 1621 * 1709

The approach we used on the first number doesn't work here.  However, notice
that these prime factors are spaced out a little.  Let's look at the indices
in the list of primes that these factors represent (13 being the 6th prime, 47
the 15th, and so on):
6, 15, 19, 22, 25, 31, 35, 42, 45, 54, 60, 63, 73, 74, 75, 79, 83, 93, 102,
108, 117, 126, 133, 142, 143, 150, 153, 158, 159, 168, 169, 178, 187, 191, 198,
203, 205, 213, 218, 222, 231, 235, 237, 246, 251, 257, 267

Note that this is a sequence of the first 267 primes where between 0-9 prime factors are
skipped at a time.  We can thus form another
number by treating the number of primes skipped as a digit:
58322536285290033985886806240808836417438318459

Once again, the prime factorization is indicative that we're on the right
track:
13217 * 14143 * 15193 * 16087 * 17011 * 18181 * 19163 * 20051 * 21143 *
22051 * 23041

At last, this gives us the information we need to produce the word.  The
first two digits of these primes are simply counting up, being used solely to
order the letters.  The last digit is chosen specifically to make the numbers
prime. The useful information lies in the third and fourth digits, which give the
value (A=1, E=5, etc.) of each letter in the answer word: UNSHARPENED